The A and F style mandolins are based on models developed by Gibson in the 1920s. They are most commonly used in bluegrass music. A Style mandolins are generally teardrop in shape and have f holes (f shaped holes like those on a violin) or an oval soundhole. The F-style Mandolin is shaped more like a guitar and has curly scroll on the top left hand side. It can also have f holes or an oval soundhole. The A and F style generally have a slightly arched top. The Flat Top or Celtic mandolin has a flat top and usually an oval soundhole and is used more in celtic music. The bowl back type of mandolin shown is mainly used in classical playing.
There are also electro-acoustic versions of mandolins. These allow you to plug the mandolin into an amp. There are pure electric versions like Eastwood's Mandocaster.
There are also other instruments in the mandolin family like the mandola and octave mandolin. See Wikipedia for more information.
When choosing a mandolin you have to try different types and see what one suits you and your budget! Things to look for are a straight neck, comfortable action (the strings are not to far from the fretboard and easy to press down) and working tuning pegs. A list of mandolin makers and retailers is on the links page.